A Comparative Study on Stabilized Finite Element Methods for the Convection-Diffusion-Reaction Problems

Abstract

The disproportionality in the problem parameters of the convection-diffusion-reaction equation may lead to the formation of layer structures in some parts of the problem domain which are difficult to resolve by the standard numerical algorithms. Therefore the use of a stabilized numerical method is inevitable. In this work, we employ and compare three classical stabilized finite element formulations, namely, the Streamline-Upwind Petrov-Galerkin (SUPG), Galerkin/Least-Squares (GLS), and Subgrid Scale (SGS) methods, and a recent Link-Cutting Bubble (LCB) strategy proposed by Brezzi and his coworkers for the numerical solution of the convection-diffusion-reaction equation, especially in the case of small diffusion. On the other hand, we also consider the pseudo residual-free bubble (PRFB) method as another alternative that is based on enlarging the finite element space by a set of appropriate enriching functions. We compare the performances of these stabilized methods on several benchmark problems. Numerical experiments show that the proposed methods are comparable and display good performance, especially in the convectiondominated regime. However, as the problem turns into reaction-dominated case, the PRFB method is slightly better than the other well-known and extensively used stabilized finite element formulations as they start to exhibit oscillations.

Investigating the spatial interaction of immune cells in colon cancer

“…To circumvent this issue, we use a space enrichment method using bubble elements. 87 , 88 , 89 This method is a very efficient stabilization method since it does not increase the size of the problem significantly. For the moved mesh we define bubble-enriched piece-wise linear function spaces , and use the mixed space given by …”

Екі Фазалы Тепе-Тең Емес Ағын Есебіндегі Қанықтықты Анықтаудың Тұрақтандырылған Ақырлы Элементтер Әдісі

“…Thus, the computational model consists of the equations( 11), (13) and the corresponding initial and boundary conditions.…”

A PDE Model of Breast Tumor Progression in MMTV-PyMT Mice

“…We enriched the usual piece-wise linear function spaces with bubble elements to remedy this. This method allows us to increase the stability without increasing the size of the problem significantly [ 128 , 129 , 130 ]. Therefore, on the moved mesh we define bubble enriched piece-wise linear function spaces , and use the following mixed space and we solve for cells and molecules concentrations in the current domain.…”